= Solution
The <symplectic neighborhood theorem> says that a symplectomorphism between closed symplectic submanifolds which lifts to an isomorphism of their <symplectic normal bundles> extends to a symplectomorphism of neighborhoods.
Let $C_X\subset X$ and $C_Y\subset Y$ be the given copies of $C$. Their self-intersection numbers are the <Euler classes> of their oriented normal bundles, so square zero makes both normal bundles trivial. The neighborhood theorem identifies neighborhoods with $C\times D^2$. Remove their interiors and identify the boundary circle bundles by a map covering the chosen identification $C_X\cong C_Y$ and reversing the normal-circle orientation. On collars, the two forms have the model
$$
\omega_C+d(r^2d\theta),
$$
and the radial coordinate can be reversed while the circle coordinate is reversed so that the forms glue. A collar application of <Moser's trick> removes any discrepancy. This proves that the <symplectic fiber sum along a square-zero surface> $X\#_CY$ has a natural symplectic form.
The displayed relation is an ordinary product relation, so
$$
\Gamma=\langle a,b,c\mid ba=ac\rangle
\cong\langle a,c\rangle=F_2,
$$
where the relation eliminates $b=aca^{-1}$. The <Gompf realization theorem> constructs a closed symplectic four-manifold $M_\Gamma$ with this fundamental group. Concretely, its construction starts from a product of a sufficiently high-genus surface and a torus, represents the two surviving generators and the relations by loops, and crosses the relevant loops with circle factors to obtain square-zero tori. Symplectic sums with copies of the <rational elliptic surface> kill the unwanted generators and impose the relations: the complement of a regular elliptic fiber is simply connected, so the <Seifert-van Kampen theorem> gives exactly $\pi_1(M_\Gamma)=\Gamma\cong F_2$.
Finally, $\mathbb{CP}^{48}$ is simply connected and has real dimension $96$. With the product symplectic form,
$$
\boxed{M_\Gamma\times\mathbb{CP}^{48}}
$$
is a closed symplectic manifold of real dimension $100$ and has fundamental group $\Gamma$.
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